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What do I do about my bad genetics that keep me from trimming down my gut?
I can only speak from my direct experience with this ... and here's the scoop: that "genetic thing" may contribute a bit, but the bottom line is that the issue is still fat, and burning fat is what will correct it.
You actually have fat all over. You might not have excessive fat on your arms or chest, for example, but a lot of your muscular definition will show as you cut body fat. You are carrying a lot of fat starting right below your chest to your waist. As you lose this fat, not only will you increase defnition, but your chest will look more muscular as it loses the fat underneath it to highlight the "cut."
You should certainly target training with weights, etc., but you still need to drop weight. You have enough fat to lose that you probably won't gain the same amount of muscle, so I would not look to bulk. I'd look to train to gain/maintain muscle and cut the fat. Once you are lean and defined, then you can decide your next step and move into a bulking phase.